Normal Distribution

Article byWallstreetmojo Team
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

What is Normal Distribution in Statistics?

A normal distribution or Gaussian distribution refers to a probability distribution where the values of a random variable are distributed symmetrically. These values are equally distributed on the left and the right side of the central tendency. Thus, a bell-shaped curve is formed.

Also, the maximum number of values appears close to the mean; the tail consists of only a few values. The empirical ruleEmpirical RuleEmpirical Rule in Statistics states that almost all (95%) of the observations in a normal distribution lie within 3 Standard Deviations from the Mean.read more applies to such probability functions. Therefore, 68% of the values lie within one standard deviation range. 95% of observations lie within two standard deviations, and 99.7% of the values appear within three standard deviations.

Key Takeaways

  • A normal distribution is a statistical phenomenon representing a symmetric bell-shaped curve. Most values are located near the mean; also, only a few appear at the left and right tails.
  • It follows the empirical rule or the 68-95-99.7 rule.
  • Here, the mean, median, and mode are equal; the mean and standard deviation of the function are 0 and 1, respectively.
  • This mathematical function has two key parameters:
    The mean (µ) and the standard deviation (σ).

Normal Distribution Explanation

A normal distribution resembles an asymmetric arrangement of most of the values around the mean, such that the curve so formed looks like a bell. It has two key parameters: the mean (µ) and the standard deviation (σ). This probability method plays a crucial role in asset return calculation and risk management strategy decisions. The following figure shows that the statistical probability function is a bell-shaped curveBell-shaped CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more that follows the empirical rule:

Normal Distribution

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The possible outcomes of the function are given in terms of whole real numbers lying between -∞ to +∞. The tails of the bell curve extend on both sides of the chart (+/-) without limits.

  1. Approximately 68% of all observations fall within +/- one standard deviation(σ).
  2. About 95% of all observations fall within +/- two standard deviations (σ).
  3. Nearly 99.7% of all observations fall within +/- three standard deviations (σ).

Skewness refers to symmetry. If skewnessSkewnessSkewness is the deviation or degree of asymmetry shown by a bell curve or the normal distribution within a given data set. If the curve shifts to the right, it is considered positive skewness, while a curve shifted to the left represents negative skewness.read more is 0, the data is perfectly symmetrical. If the normal distribution is uneven with a skewness greater than zero or positive skewness, then its right tail will be more prolonged than the left. Similarly, for negative skewnessNegative SkewnessThe negatively skewed distribution is one in which the tail of the distribution is longer on the left side and more values are plotted on the right side of the graph. Due to the negative distribution of data, the mean is lower than the median and mode.read more, the left tail will be longer than the right tail. Negative skewness means skewness is less than zero.

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KurtosisKurtosisKurtosis in statistics is used to describe the distribution of the data set and depicts to what extent the data set points of a particular distribution differ from the data of a normal distribution. It determines whether the data is heavy-tailed or light-tailed.read more is a measure of peakiness. If the kurtosis is 3, the probability data is neither too peaked nor too thin at tails. If the kurtosis is more than three, then the data curve is heightened with fatter tails. Alternatively, if the kurtosis is less than three, then the represented data has thin tails with the peak point lower than the normal distribution. For a normal distribution, the kurtosis is 3.

Kurtosis

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Characteristics of Normal Distribution

Normal Distribution has the following characteristics that distinguish it from the other forms of probability representations:

  • Empirical Rule: In a normal distribution, 68% of the observations are confined within -/+ one standard deviation, 95% of the values fall within -/+ two standard deviations, and almost 99.7% of values are confined to -/+ three standard deviations.
  • Bell-shaped Curve: Most of the values lie at the center, and fewer values lie at the tail extremities. This results in a bell-shaped curve.
  • Mean and Standard Deviation: This data representation is shaped by mean and standard deviation.
  • Equal Central Tendencies: The mean, median, and mode of this data are equal.
  • Symmetric: The normal distribution curve is centrally symmetric. Therefore, half of the values are to the left of the center, and the remaining values appear on the right. 
  • Skewness and Kurtosis: Skewness is the the symmetry. The skewness for a normal distribution is zero. Kurtosis studies the tail of the represented data. For a normal distribution, the kurtosis is 3.
  • Total Area = 1: The total value of the standard deviation, i.e., the complete area of the curve under this probability function, is one. Also, the entire mean is zero.

Normal Distribution Curve

The curve takes the shape of a bell due to the symmetrical arrangement of the values that are concentrated towards the central tendencyCentral TendencyCentral Tendency is a statistical measure that displays the centre point of the entire Data Distribution & you can find it using 3 different measures, i.e., Mean, Median, & Mode.read more. At the same time, the tail consists of an insignificant number of values.

Have a look at the curve below to understand its shape better:

Normal Distribution Curve

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Normal Distribution Formula

The Probability Density Function (PDF) of a random variable (X) is given by:

Normal Distribution Formula

Where;

  • -∞ < x < ∞; -∞ < µ < ∞; σ > 0
  • F(x) = Normal probability Function
  • x = Random variable
  • µ = Mean of distribution
  • σ = Standard deviation
  • of the distribution
  • π = 3.14159
  • e = 2.71828

Transformation (Z)

When it comes to a comparative study of two or more samples, there arises a need for converting their values in z-scores. This is termed as z-transform.

Transformation Formula

For ascertaining the z-score, the following formula is used:

Transformation Formula

Where X = Random variable.

Normal Distribution Table

The table referred for the standard deviation is the z-table. Here, we determine the probability of getting a particular outcome using the transformation formula to ascertain the value of the z-score, which is depicted in percentage using a z-table.

Examples

Example #1

Let us suppose that a company has 10000 employees and multiple salary structures according to specific job roles. The salaries are generally distributed with the population meanPopulation MeanThe population mean is the mean or average of all values in the given population and is calculated by the sum of all values in population denoted by the summation of X divided by the number of values in population which is denoted by N.read more of µ = $60,000, and the population standard deviation σ = $15000. What will be the probability of a randomly selected employee earning less than $45000 per annum?

Normal Distribution Examples

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Solution:

As shown in the above figure, we need to find out the area under the normal curve from 45 to the left side tail to answer this question. Also, we need to use the z-table value to get the correct answer.

Firstly, we need to convert the given mean and standard deviationStandard DeviationStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability.read more into a standard normal distribution with mean (µ)= 0 and standard deviation (σ) =1 using the transformation formula.

After the conversion, we need to look up the z-table to find out the corresponding value, which will give us the correct answer.

Given,

  • Mean (µ) = $60,000
  • Standard deviation (σ) = $15000
  • Random Variable (x) = $45000

Transformation (z) = (45000 – 60000 / 15000)

Transformation (z) = -1

z-table

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The value equivalent to -1 in the z-table is 0.1587, representing the area under the curve from 45 to the left. Thus, it indicated that when we randomly select an employee, the probability of making less than $45000 a year is 15.87%.

It is important to note that we have converted the z-score value 0.1587 into a percentage by multiplying it by 100 to get 15.87%.

Example #2

For the same above scenario, now find the probability of a randomly selected employee earning more than $85,000 a year.

Normal Distribution Examples 1

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Solution:

So, in this question, we need to find out the shaded area from 85 to right tail using the same formula.

Given:

  • Mean (µ) = $60,000
  • Standard deviation (σ) = $15000
  • Random Variable (X) = $85,000

Transformation (z) = (85000 – 60000 /15000)

Transformation (z) = 1.67

As per the Z-table, the equivalent value of 1.67 is 0.9525 or 95.25%, which shows that the probability of randomly selecting an employee earning less than $85,000 per annum is 95.25%.

But as per the question, we need to determine the probability of random employees earning more than $85,000 a year, so we need to subtract the calculated value from 100.

  • Random Variable (X) = 100% – 95.25%
  • Random Variable (X) = 4.75%

So, the probability that employees earn more than $85,000 per year is 4.75%.

Uses

This mathematical function is applied in various fields of study, whether it is science, economicsEconomicsEconomics is an area of social science that studies the production, distribution, and consumption of limited resources within a society.read more, statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more, finance, business, investment, psychology, health, genetics, biotech, or academics. Some of its typical applications are discussed below:

The Gaussian Function is commonly used in data science and data analytics. Advanced technologies like artificial intelligence (AI) and machine learning can deliver better results when used along with normal density functions.

Frequently Asked Questions (FAQs)

Why is normal distribution important?

This Gaussian function is one of the most popular probability density functions. This is because it efficiently provides the close-by results or probability to natural phenomena. Thus, it is universally applied in numerous fields like economics, finance, investment, psychology, science, health, business, and economics.

What to do if data is not normally distributed?

One can check for data-entry errors, measurement errors, and outliers in case of a skewed or non-normal distribution. After that, these issues can be reviewed to eliminate errors and normalize the represented data.

How do you know if a distribution is normal?

If the following conditions are fulfilled, then the represented data is normal:
• Centrally symmetric and bell-shaped curve,
• Equal Mean, Median, and Mode,
• The distribution Mean is 0,
• The standard deviation is 1.
• Skewness is 0. and
• Kurtosis is 3.s

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